Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 8 + \left(13 a + 6\right)\cdot 17 + \left(8 a + 5\right)\cdot 17^{2} + 2 a\cdot 17^{3} + \left(12 a + 12\right)\cdot 17^{4} + \left(9 a + 12\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 5 + \left(a + 5\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(6 a + 5\right)\cdot 17^{3} + \left(16 a + 10\right)\cdot 17^{4} + a\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 16 + \left(3 a + 11\right)\cdot 17 + 8 a\cdot 17^{2} + \left(14 a + 11\right)\cdot 17^{3} + \left(4 a + 4\right)\cdot 17^{4} + \left(7 a + 10\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 11\cdot 17 + 8\cdot 17^{2} + 4\cdot 17^{4} + 7\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 13 + \left(15 a + 15\right)\cdot 17 + \left(3 a + 6\right)\cdot 17^{2} + \left(10 a + 16\right)\cdot 17^{3} + 2\cdot 17^{4} + \left(15 a + 3\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
